This was a mistake. You are right, gfortran seed is 8 integers, or 256 bits. I edited my original post to correct this mistake.
BTW, the NAG compiler seed is 630 integers. Each compiler is free to choose its own algorithm, which typically requires different numbers of bits of internal state storage.
In a parallel environment, this is done with the IMAGE_DISTINCT argument to the RANDOM_INIT() intrinsic as described here (RANDOM_INIT (The GNU Fortran Compiler)) for gfortran. This uses the jump algorithm mentioned in the wikipedia page to skip forward/backward. This is an inexpensive operation. In a single thread environment, I don’t think the programmer can jump easily. That might be a handy thing to be able to do.
I thought the wikipedia page had a good reference for the jump operation, but I’m not seeing it now. I did a google search, and I also don’t see a good reference for this in the first page. I know it involves computing a polynomial with modular arithmetic that depends on the number of jump steps, and then operating on the 256 bits of state, but I don’t have the original reference handy where I read this.